Suppose `G`

be a planar graph with `16`

vertex, that degree of each vertex is `4`

.

When we place this graph on plane as a planar graph, each finite region be a trilateral or foursquare and boundary of infinite region is made up by `4`

edges.

How many trilateral edge be in such graph? my TA solve and find the triangle is `8`

by drawing a figure. Any other idea to calculate it ?

Answer:

Let V be the number of vertices, E be the number of edges, F3 be the number of triangles, and F4 be the number of quads. We have equations

```
V = 16 # given
4 V = 2 E # number of vertex-edge incidences
3 F3 + 4 F4 = 2 E # number of face-edge incidences
V - E + F3 + F4 = 2 # Euler's polyhedron formula,
```

which suffice to derive that V = 16, E = 32, F3 = 8, F4 = 10.

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